16 



GENERALIZED THEORY OF ELECTROACOUSTIC TRANSDUCERS 



37 MAXIMUM ELECTRIC POWER 



OUTPUT ON RECEPTION AND 

 THRESHOLD PRESSURE 



Maximum electric power P max is transferred to a 

 receiver when the electric load impedance of the 

 receiver is the complex conjugate of the effective 

 electric impedance of the transducer. Under this con- 

 dition, by Thevenin's theorem 



p = tH- 



1 max i) 



F - F ' - , 



R o1 4R„ 



|jb inc l 2 Af 2 _lfr 



(35) 



4R„ 



P c 4tt8 "' 



S-8 GENERALIZATION OF THEORY TO 

 ANY TYPE OF DIAPHRAGM 

 MOTION 



The treatment of the theory of electroacoustic 

 transducers given above for transducers in which the 

 diaphragm velocity is the same at all points can 

 readily be generalized to the case where there is no 

 such restriction on the velocity distribution. The 

 form that the generalization takes follows in outline. 

 The fundamental equations for the transducer can 

 be written as 



p(r) = / z (r,r') v n (r') dr' + k(r) I (37) 



where E s is the signal voltage across the load in the 

 matched circuit, E s[oc) is the signal voltage that woidd 

 be developed by the transducer on open circuit, M 

 is the receiving sensitivity, and E p is the projector 

 efficiency. The last form of equation (35) follows 

 from the last form of equation (31). Equation (35) 



suggests that the quantity -A—, which has the dimen- 



47r8 

 sions of an area and is usually called the effective 



area, has the significance of being the maximum 



cross section for energy absorption by the transducer 



from the sound field. This follows from the fact that 



and 



lis the incident intensity of sound, E p never ex- 



\P 2 UH 



ceeds unity, and maximum power is absorbed in a 

 matched circuit. 



An important parameter of the transducer is its 

 threshold pressure p t . This is defined as the pressure 

 in a uniform, plane-wave, free sound field propagated 

 parallel to the acoustic axis of the transducer, which 

 produces a signal power output in the load equal to 

 the inherent thermal noise power in the load. (See 

 Chapter 4 for a full discussion.) The noise power is 

 taken in a 1-cycle band and the transducer is sup- 

 posed to be in a matched circuit. The noise power in 

 the load in a matched circuit is one-half of the open- 

 circuit noise power in the transducer (since noise 

 pressures add in random phase). This open-circuit 

 noise power in a 1-cycle band is given by 4KT where 

 K is Boltzmann's constant and T is the absolute tem- 

 perature of the device. 70 Consequently, the threshold 

 pressure is given by the relation 



P c 4^8 p 2 ( h 



(36) 



E = / A''(r') v„(r') dr' + Z„ L 



(38) 



Here r and >' are points on the transducer surface S; 

 /j(r) and v n (i) are the pressure and normal velocity at 

 the point r, respectively. The functions z (r,r'), k(r), 

 and A'(r') are functions characteristic of the trans- 

 ducer which are the generalizations of z , k, and k' in 

 the simpler treatment given earlier. For coupling to 

 an electric source of generated voltage £,, and inter- 

 nal impedance Z Ult , the equation 



/• = £n 



Zin. / 



(4) 



again holds. However, for coupling on the acoustic 

 side we now have 



p(R) = p QL) - j z r (K, r') v„(r') dr' (39) 



where 



Po(R) = p iM (R) - if^P r ' (R ' r ' } dr ' 



and 



z,.(r,r')= 7 <;(,-,,-) 



(40) 

 (41) 



and R is any point in the medium. (R may be taken 

 equal to r.) Here G(r,r') is the same Green's function 

 introduced earlier. The quantity z,.(r,r') is an acoustic 

 radiation impedance continuous matrix which is the 



